Block maxima
Peaks over threshold
Largest in a block of observations
Distribution over a threshold
\[\{X_1, X_2, ..., X_n \}\]
\[M_n = \text{max}\{X_1, X_2, ..., X_n \}\]
If \(\text{Pr}(M_n \leq z)\) is well-behaved, it belongs to one of 3 families:
\[G(z) = \text{exp}\Big\{- \Big[ 1 + \xi \Big( \dfrac{z - \mu}{\sigma} \Big) \Big]^{-1/\xi} \Big\}\]
defined over \(\{z: 1 + \xi (z - \mu)/\sigma \gt 0 \}\)
\[G(z) = \text{exp}\Big\{- \Big[ 1 + \xi \Big( \dfrac{z - \mu}{\sigma} \Big) \Big]^{-1/\xi} \Big\}\]
The shape parameter determines the GEV family
Observe maxima
Estimate GEV parameters
Infer probabilities, future extremes, etc.
Observe Boulder Creek discharge data
Estimate GEV parameters
Infer return intervals for 2013 floods